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MAT 151: College Algebra with Trigonometry

Subsection 2.5 Coordinating Period Amplitude and Midline Exercises

  1. Refer to Figure 2.5.16.
    A graph on a coordinate grid showing a periodic function with a maximum value at the point (0, 1) and a minimum value at the point (4, −5). The curve decreases from the maximum at x = 0 to the minimum at x = 4.
    Figure 2.5.16.
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    1. Write a function equation for \(f\) of the form \(f(x)=a \sin (b(x-c))+d\text{.}\)
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    2. Write a function equation for \(f\) of the form \(f(x)=a \cos (b(x-c))+d\text{.}\)
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  2. Predict what each of the following graphs will look like and sketch. Use Desmos to check your answer. Be sure to label the amplitude, period, midline and endpoints.
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    1. \(y=3\sin (\pi (x-1))-2\) on \(-4 \leq x \leq 4\)
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    2. \(y=\frac{1}{3}\cos (\frac{\pi}{2} (x+1))\) on \(-8 \leq x \leq 8\)
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    3. \(y=\frac{1}{4}\sin(\frac{1}{2}(x-2))-3\) on \(0 \leq x \leq 26\)
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    4. \(y=-5\cos (\frac{1}{\pi} x)+5\) on \(-10 \leq x \leq 10\)
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    5. \(y=4\sin(5\pi x)+2\) on \(0 \leq x \leq 6\)
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    6. \(y=15\cos(\frac{2\pi}{3}(x-\pi))+35\) on \(-6 \leq x \leq 6\)
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  3. Refer to the graph of \(f\) in Figure 2.5.17.
    A rapidly oscillating trigonometric graph with many closely spaced peaks and troughs over a short horizontal interval. There is a minimum at (0,1) and another minimum at (1,1). There is a maximum at (0.5, 9).
    A trigonometric graph with moderate frequency showing repeating peaks and valleys across the interval from about x = -4 to x = 3. There is a maximum at (0,2) and a minimum at (1,-8).
    Figure 2.5.17.
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    1. Write a function equation for \(f\) of the form \(a \sin (b(x-c))+d\text{.}\)
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    2. Write a function equation for \(f\) of the form \(a \cos (b(x-c))+d\text{.}\)
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  4. Refer to the graph of \(g\) in Figure 2.5.17.
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    1. Write a function equation for \(g\) of the form \(a \sin (b(x-c))+d\text{.}\)
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    2. Write a function equation for \(g\) of the form \(a \cos (b(x-c))+d\text{.}\)
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  5. The population of Butler has been declining, as seen in Table 2.5.18.
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    Table 2.5.18. Population of Butler
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    \(t\) (years since 1990) 0 \(5\) 10 \(15\) 20
    P(t) (people) 89,480 \(80,061\) \(71,633\) \(64,093\) \(57,346\)
    1. Decide what kind of function (linear, quadratic, exponential) is appropriate to model the population of Butler, and explain your reasoning.
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    2. Write a function equation for \(P(t)\text{.}\)
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    3. According to your function equation, what was the population of Butler in 2015?
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    4. What was the average annual decline in the population of Butler from 1990 to 2010?
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    5. When will the population of Butler decline to half of what it was in 1990?
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    6. The population of nearby Greenville is described by the equation \(G(t)=2P(t)\text{.}\) Compare the population of Greenville with the population of Butler.
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    7. Write an equation for \(P^{-1}\text{.}\) State the units of the domain and range of \(P^{-1}\text{.}\)
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  6. Write as a single logarithm.
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    1. \(\displaystyle \log_3 y - 4 \log_3 2x\)
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    2. \(\displaystyle 2(\ln x-\ln 3y)-(\ln z+2\ln y)\)
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    3. \(\displaystyle \log x \cdot \log 2\)
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  7. Perform the indicated operation and simplify if possible.
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    1. \(\displaystyle \frac{\frac{x+3}{x^2-25}}{\frac{x^2-9}{x+5}}\)
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    2. \(\displaystyle \frac{5x+5y}{y-x} \div \frac{15}{3x-3y}\)
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    3. \(\displaystyle \frac{t}{t^2-t-30}-\frac{1}{t+5}\)
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